Forty-six percent of those polled said they would vote for Ms. Grimes, who has campaigned primarily on bringing jobs to Kentucky and raising the minimum wage, while just 42 percent said they would vote for Mr. McConnell, the Courier-Journal reported.

“When he says that raising the minimum wage is the last thing he wants to do, Mitch McConnell apparently thinks that over 60 percent of Kentuckians are wrong. Only 32 percent oppose raising the minimum wage — a number that matches McConnell’s dismal approval rating here in the Commonwealth,” Charly Norton, a spokeswoman for Ms. Grimes said in a statement on Monday. “Mitch McConnell is clearly out of touch, out of ideas and out of time.”

Allison Moore, a spokeswoman for Mr. McConnell, said his re-election team is not worried by the early poll numbers.

“We’re very comfortable about where this race stands and extremely confident that Senator McConnell will earn the votes of Kentuckians this fall,” she said. “The contrast between Mitch McConnell’s conservative accomplishments for Kentucky and Alison Lundergan Grimes’s alliance with President Obama’s agenda of Obamacare and the war on coal will become very clear to everyone over the next nine months.”

Despite Mr. McConnell’s low job approval rating in the state, he doesn’t seem to face a real threat from his tea party challenger Matt Bevin, who trails Mr. McConnell by 26 points, according to the local newspaper.